Question

If $\alpha ,\beta$ are the roots of $x^2-x+2=0$, then ${\alpha }^3\beta +\alpha {\beta }^3$

• A. $-6$
• B. $6$
• C. $-10$
• D. $-3$

Answer 1

The correct option is A $-6$

$\Rightarrow$ The given quadratic equation is $x^2-x+2=0$

$\Rightarrow$ Roots are $\alpha$ and $\beta$.

$\Rightarrow$ Product of the roots $=\frac{c}{a}$

$\therefore$ $\alpha \beta =\frac{2}{1}=2$ ------ ( 1 )

$\Rightarrow$ Sum of the roots $=\frac{-b}{a}$

$\therefore$ $\alpha +\beta =\frac{-(-1)}{1}$

$\Rightarrow$ $\alpha +\beta =1$

$\Rightarrow$ $(\alpha +\beta {)}^2=(1{)}^2$ [ Squaring both the sides ]

$\Rightarrow$ ${\alpha }^2+2\alpha \beta +{\beta }^2=1$

$\Rightarrow$ ${\alpha }^2+{\beta }^2=1-2\alpha \beta$

$\Rightarrow$ ${\alpha }^2+{\beta }^2=1-2\left(2\right)$ [ From ( 1 ) ]

$\therefore$ ${\alpha }^2+{\beta }^2=-3$ ------ ( 2 )

$\Rightarrow$ Now, $a^3\beta +\alpha {\beta }^3$

$\Rightarrow$ $\alpha \beta ({\alpha }^2+{\beta }^2)$

$\Rightarrow$ $\left(2\right)(-3)$ [ From ( 1 ) and ( 2 ) ]

$\Rightarrow$ $-6$